IVORY'S THEOREM. 



<u ike product of the other two O2f of the fir* 

 'id t* to th* correiptmding product in the Momd MptM: 



thf tir<> r // til*" i</j bfimi MMMMM0MH <iml <>f t/tf .'///< f/-/Mi''y. 



Let M semi-axes of the first ellipsoid; a', 



thaw of the second. !/ t /,.</, A) denote a point on the sur- 

 face of t .-: corresponding point 

 on the surface of the second cllijw- 



attraction of the first ellipsoid on a particle at (f\g> h') 

 resolved parallel to the axis of 



///. 



^.( a ,-/r+( y -^r+(*-*r, 



iw of attraction is represented bv (r) ; /i is a con- 

 stant : the integration is to extend throughout the volume of 

 the first ellipsoid. 



Let J>(r)efr-^(r\ Integrate with respect to x; and 

 let r an the values of r at the extremities of a 



< lli[>soid parallel to the axis of x. Thus the 

 resolved attractioi 



same way the resolved attraction of the second 

 >id on the corresponding point on the surface of the 

 ellipsoid may be expressed by 



Now suppose that we always make 



then we have by the preliminary proposition 

 r.-r, and r t -r, ; 



and we have also 



dyd* be 

 '"' 



rst resolved attraction is to the second as be is 

 to '' establishes the theorem. 



